Interpolated

Interpolated strain plains to ACI318 and similar codes

ACI318 and several other codes give a method to compute a value of the second moment of area intermediate between that of the uncracked, $I_g$, and fully cracked, $I_{c r}$, values, using the following expression:

$$I_e=\left(\frac{M_{c r}}{M_a}\right)^3 I_g+\left[1-\left(\frac{M_{c r}}{M_a}\right)^3\right] I_{c r}$$

where $M_{c r}$ is the cracking moment and $M_a$ is the applied moment.

AdSec SLS analyses determine a strain plane intermediate to the uncracked and fully cracked strain planes. The program determines a value for $\zeta$, the proportion of the fully cracked strain plane to add to the proportion $(1-\zeta)$ of the uncracked plane so that the resulting plane is compatible with ACI318's approach. Unfortunately, since ACI318's expression is an interpolation of the inverse of the curvatures, rather than the curvatures themselves, there is no direct conversion. It should also be noted that although $I_g$ is defined as the value of second moment of area ignoring the reinforcement, it is assumed that this definition was made for simplicity, an: AdSec includes the reinforcement.

Let $\alpha=\left(M_{c r} / M_a\right)^3$, the uncracked curvature be $\kappa_I$ and the fully cracked curvature be $\kappa_{II}$.

To ACI318, the interpolated curvature

$$\kappa=\frac{1}{\left[\alpha / \kappa_I+(1-\alpha) / \kappa_{II}\right]},$$

and the aim is to make this equivalent to

$$\kappa=\zeta \kappa_{I I}+(1-\zeta) \kappa_I,$$

Equating these two expressions gives

$$\alpha \zeta \kappa_{\Pi} / \kappa_I+\alpha(1-\zeta)+(1-\alpha) \zeta+(1-\alpha)(1-\zeta) \kappa_I / \kappa_{II}=1$$

which can be re-arranged to give

$$\zeta=\frac{1}{\left[1+\left(\kappa_{II} / \kappa_I\right) /(1 / \alpha-1)\right]}$$

The ratio $\kappa_{II} / \kappa_I$ is appropriate for uniaxial bending. For applied loads $\left(N, M_y, M_z\right)$, and uncracked and fully cracked strain planes $\left(\varepsilon_I, \kappa_{y I}, \kappa_{z I}\right)$ and $\left(\varepsilon_{I I}, \kappa_{y II}, \kappa_{z I I}\right)$ respectively, $\kappa_{I I} / \kappa_I$ is replaced by the ratio $\left(N \varepsilon_{I I}+M_y \kappa_{y I I}++M_z \kappa_{z I I}\right) /\left(N \varepsilon_I+M_y \kappa_{j I}++M_z \kappa_{z I}\right)$, which is independent of the location chosen for the reference point. In the absence of axial loads, this ratio ensures that the curvature about the same axis as the applied moment will comply with ACI318; in the absence of moments, the axial strain will follow a relationship equivalent to that in ACI318 but using axial stiffness as imposed to flexural stiffness.

The ratio $\left(M_{c r} / M_a\right)$ is also inappropriate for general loading. For the general case, it is replaced by the ratio $\left(f_{c t} / \sigma_{t l}\right)$, where $f_{c t}$ is the tensile strength of the concrete and $\sigma_{t l}$ is the maximum concrete tensile stress on the uncracked section under applied loads.

Summary:

$$\zeta=1 /\left[1+\frac{\left(N \varepsilon_{I I}+M_y \kappa_{y I I}+M_z \kappa_{z I I}\right) /\left(N \varepsilon_I+M_y \kappa_{y I}+M_z \kappa_{z I}\right)}{\left(\sigma_{t I} / f_{c t}\right)^3-1}\right]$$

Since $\zeta$ is larger for short-term loading, all curvatures and strains are calculated based on short-term properties regardless of whether $\zeta$ is subsequently used in a long-term serviceability calculation.